I teach an introductory course on basic electronics and instrumentation. I recently put together a document on using Lissajous figures to experimentally determine the phase shift of an LTI system. I thought someone here might find it useful:
It's a short document (2 pages). I included a couple of graphics that I think could be helpful.
*) The graphic on the first page shows how to use the shape and rotation direction of the Lissajous figure to properly place the phase shift (i.e., determine which quadrant the angle is in).
*) The graphic on the second page gives a simple laboratory procedure for determining phase shift (including steps that put the phase shift in the correct quadrant).